Heat vegetable oil over medium-high heat in a medium-sized pan. Cook and stir ground chicken in the hot skillet until browned and crumbly, 5 to 7 minutes. Drain and discard grease.
Combine chili powder, garlic, cumin, and red pepper flakes in a small bowl; stir into chicken to coat.
Transfer chicken to a slow cooker. Add kidney beans, diced tomatoes with chiles, black beans, onion, and chopped jalapeno. Pour chicken broth over mixture and stir to combine.
Cover and cook on High until chili has cooked down and thickened, about 4 hours, or on Low for about 8 hours. Stir corn kernels into mixture during last 30 minutes of cook time.
Spoon chili into bowls and top each with sour cream, Cheddar cheese, green onions, and sliced jalapeno.
